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Algebraic and Geometric Surgery$

Andrew Ranicki

Print publication date: 2002

Print ISBN-13: 9780198509240

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198509240.001.0001

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THE ODD-DIMENSIONAL SURGERY OBSTRUCTION

THE ODD-DIMENSIONAL SURGERY OBSTRUCTION

Chapter:
(p.302) 12 THE ODD-DIMENSIONAL SURGERY OBSTRUCTION
Source:
Algebraic and Geometric Surgery
Author(s):

Andrew Ranicki

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198509240.003.0012

Abstract and Keywords

This chapter provides the algebraic construction and geometric properties of the odd-dimensional surgery obstruction groups. It includes quadratic and kernel forms.

Keywords:   quadratic formation, kernel formation, odd-dimensional L-group, odd-dimensional surgery obstruction, linking form

This chapter defines the Wall surgery obstruction of a (2n+1)-dimensional degree 1 map (f, b) : MX,

σ * ( f , b ) L 2 n + 1 ( [ π 1 ( X ) ] ) .
The main result is that σ *(f, b) = 0 if (and for n ⩾ 2 only if) (f, b) is normal bordant to a homotopy equivalence.

Section 12.1 introduces the notion of an ε-quadratic formation (K, λ, μ; F, G), which is a nonsingular ε-quadratic form (K, λ, μ) with lagrangians F, G. Section 12.2 constructs a kernel (−1)n-quadratic formation over Z[π1(X)] for an n-connected (2n + 1)-dimensional degree 1 normal map (f, b). The (2n + 1)-dimensional surgery obstruction group L 2n+1(A) of cobordism classes of nonsingular (−1)n-quadratic formations (K, λ, μ; F, G) over A is defined in Section 12.3. The odd-dimensional surgery obstruction is defined in Section 12.4 to be the cobordism class of a kernel formation. Section 12.5 describes the algebraic effect on a kernel formation of a geometric surgery on (f, b). Finally, Section 12.6 gives a brief account of linking forms, the original approach to odd-dimensional surgery obstruction theory.

Odd-dimensional surgery obstruction theory is technically more complicated than the even-dimensional case. Specifically, an n-connected 2n-dimensional normal map has a unique kernel form, whereas an n-connected (2n + 1)-dimensional normal map has many kernel formations.

12.1 Quadratic formations

The surgery obstruction of an even-dimensional degree 1 normal map was expressed in Section 11.5 as the equivalence class of the kernel nonsingular ε-quadratic form, with the zero class represented by the forms which admit a lagrangian. The different lagrangians admitted by the kernel form correspond to different ways of solving the even-dimensional surgery problem by a normal bordism to a homotopy equivalence. The odd-dimensional surgery obstruction will be expressed in Section 12.4 as an equivalence class of ε-quadratic ‘formations’, which are nonsingular ε-quadratic forms with ordered pairs of lagrangians, corresponding to two solutions of an even-dimensional surgery problem in codimension 1.

(p.303) The fundamental algebraic structure determined by a closed (n − 1)-connected 2n-dimensional manifold N 2n is the nonsingular (−1)n-symmetric form (H n(N),λ) over Z. The fundamental algebraic structure determined by an (n − 1)-connected (2n + 1)-dimensional manifold with boundary (M 2n+1,∂M) with H n(M, ∂M) = 0 is the lagrangian of the (−1)n-symmetric form (H n(∂M),λ) defined by

L = im ( H n + 1 ( M , M ) H n ( M ) ) H n ( M ) .
Now suppose that M is a closed (2n+1)-dimensional manifold which is expressed as a union of two (2n+1)-dimensional manifolds ( M + 2 n + 1 , M + ) , ( M 2 n + 1 , M ) with the same (n − 1)-connected boundary N 2n = ∂M + = ∂M -:
                   THE ODD-DIMENSIONAL SURGERY OBSTRUCTION
If M, M +, M -, N are (n − 1)-connected and H n(M +, N) = H n(M -, N) = 0 the (−1)n-symmetric form on K = H n(N) has lagrangians
L ± = im ( H n + 1 ( M ± , N ) H n ( N ) ) ,
giving a(-1)n - symmetric formation (K,λ;L+L-). Such splitings were first used by Heegaard (in 1898) in the study of 3-diamensional manifolds: every conected Mainfold M3 can be expressed as a union,
M = M N M + ,
of solid tori along a genus g surface, so that
M + = M = # g S 1 × D 2 , M + M = N = M ( g ) = # g S 1 × S 1 ,
corresponding to a handle decomposition
M = ( h 0 g h 1 ) ( g h 2 h 3 ) .
(p.304) It should be clear that such expressions are not unique, since forming the connected sum of M with
S 3 = ( D 2 × D 2 ) = S 1 × D 2 D 2 × S 1
increases the genus g by 1 without affecting the diffeomorphism type.

The remainder of this section will only consider the algebraic properties of formations. As before, let A be a ring with involution, and let ε = ±1.

Definition 12.1 An ε-quadratic formation over A (K, λ, μ; F, G) is a nonsingular ε-quadratic form (K, λ, μ) together with an ordered pair of lagrangians F, G. □

Strictly speaking, Definition 12.1 defines a ‘nonsingular formation’. In the general theory a formation (K, λ, μ; F, G) is a nonsingular form (K, λ, μ) together with a lagrangian F and a sublagrangian G, with F, G, and K f.g. projective. For basic odd-dimensional surgery obstruction theory only nonsingular formations with F, G, and K f.g. free need be considered. Also, A can be assumed to be such that the rank of f.g. free A-modules is well-defined (e.g. A = Z[π]), with A k isomorphic to A l if and only if k = l. This hypothesis ensures that for every formation (K, λ, μ; F, G) over A there exists an automorphism a: (K, λ, μ) → (K, λ, μ) such that α(F) = G (Proposition 12.3 below). In the original work of Wall [92, Chapter 6] the odd-dimensional surgery obstruction was defined in terms of such automorphisms. Novikov [64] proposed the use of pairs of lagrangians instead, and the expression of odd-dimensional surgery obstructions in terms of formations was worked out in Ranicki [68].

Definition 12.2 An isomorphism of ε-quadratic formations over A,

f : ( K , λ , μ ; F , G ) ( K , λ , μ , F , G ) ,
is an isomorphism of forms f: (K, λ, μ) → (K′, λ′, μ′) such that
f ( F ) = F , f ( G ) = G .
 □

Proposition 12.3

  1. (i) Every ε-quadratic formation (K, λ, μ; F, G) is isomorphic to one of the type (H ε(F); F, G).

  2. (ii) Every ∈-quadratic formation (K, λ, μ; F, G) is isomorphic to one of the type (H (F); F, α(F)) for some automorphism α: H ε(F) → H ε(F).

Proof (i) By Theorem 11.51 the inclusion of the lagrangian FK extends to an isomorphism of ε-quadratic forms,

f : H ( F ) ( K , λ , μ ) ,
defining an isomorphism of ε-quadratic formations,
f : ( H ( F ) ; F , f 1 ( G ) ) ( K , λ , μ ; F , G ) .

(p.305) (ii) As in (i) extend the inclusions of the lagrangians F → K, G → K to isomorphisms of forms

f : H ( F ) ( K , λ , μ ) , g : H ( G ) ( K , λ , μ ) .
Then,
rank A ( F ) = rank A ( K ) / 2 = rank A ( C ) ,
so that F is isomorphic to G. Choosing an A-module isomorphism β: F → G there is defined an automorphism of H (F)
α : H ( F ) ( β 0 0 β 1 ) H ( G ) g ( K , λ , μ ) f 1 H ( F )
such that there is defined an isomorphism of formations
f : ( H ( F ) ; F , α ( F ) ) ( K , λ , μ ; F , G ) .
 □

In Section 12.2 there is associated to an n-connected (2n + 1)-dimensional degree 1 normal map (f, b): M 2n+1X a ‘stable isomorphism’ class of kernel (−1)n-quadratic formations (K, λ, μ; F, G) over Z[π1(X)] such that

K n ( M ) = K / ( F + G ) , K n + 1 ( M ) = F G .
Stable isomorphism is defined as follows:

Definition 12.4

  1. (i) An ε-quadratic formation T = (K, λ, μ; F, G) is trivial if it is isomorphic to (H ε(F); F, F*).

  2. (ii) A stable isomorphism of ε-quadratic formations,

    [ f ] : ( K , λ , μ : ˙ F , G ) ( K , λ , μ : ˙ F , G ) ,
    is an isomorphism of ε-quadratic formations of the type
    f : ( K , λ , μ : ˙ F , G ) T ( K , λ , μ : ˙ F , G ) T ,
    with T, T′ trivial. □

Proposition 12.5

  1. (i) An ε-quadratic formation (K, λ, μ; F, G) is trivial if and only if the lagrangians F and G are direct complements in K:

    F G = { 0 } , F + G = K .

  2. (p.306)
  3. (ii) A stable isomorphism [f]: (K, λ, μ; F, G) → (K′, λ′, μ′; F′, G′) of ε-quadratic formations over A induces A-module isomorphisms

    F G F G , K / ( F + G ) K / ( F + G ) .

  4. (iii) For any ε-quadratic formation (K, λ, μ; F, G) there is defined a stable isomorphism

    [ f ] : ( K , λ , μ ; G , F ) ( K , λ , μ ; F , G ) .

Proof (i) If F and G are direct complements in K, represent (λ, μ) by a split ε-quadratic form (K, ψ ∈ S(K)) with

ψ = ( α α * γ δ β β * ) : K = F G K * = F * G * .
Then, γ + εδ* ∈ HomA(G, F *) is an A-module isomorphism, and there is defined an isomorphism of formations
( 1 0 0 ( γ + δ * ) 1 ) : ( H ( F ) ; F , F ) ( K , λ , μ ; F , G ) .

(ii) By (i) an ε-quadratic formation (K, λ, μ; F, G) is trivial if and only if

F G = 0 , K / ( F + G ) = 0.

(iii) By Proposition 12.3 we can take

( K , λ , μ ; G ) = ( H ( F ) ; F , α ( G ) ) ,
with
α = ( γ δ ˜ δ γ ˜ ) : H ( G ) H ( F )
an isomorphism of hyperbolic ε-quadratic forms, which defines an isomorphism of ε-quadratic formations
α : ( H ( G ) ; G , α 1 ( F ) ) ( K , λ , μ ; G , F ) .
The A-module isomorphism,
f = ( γ δ ˜ 0 γ δ γ ˜ γ ˜ 0 0 0 γ ˜ δ 0 0 δ ˜ γ ) : G G * G * G F F * F * F ,
defines an isomorphism of ε-quadratic formations,
f : ( H ( G ) ; G , α 1 ( F ) ) ( H ( G * ) ; G * , G ) ( H ( F ) ; F , α ( G ) ) ( H ( F * ) ; F * , F ) .
giving a stable isomorphism
[ f ] : ( K , λ , μ ; G , F ) ( K , λ , μ ; F , G ) .
 □

(p.307) 12.2 The kernel formation

An n-connected (2n+1)-dimensional degree 1 normal map (f, b): M 2n+1X is such that K i(M) = 0 for in, n + 1. We shall now construct a stable isomorphism class of kernel (−1)n-quadratic formations (K, λ, μ; F, G) over Z[π1(X)] such that

K n ( M ) = K / ( F + G ) , K n + 1 ( M ) = F G ,
using the following generalization of the Heegaard splitting of a 3-dimensional manifold as a union of solid tori.

The kernel Z[π1 (X)]-module K n (M) is finitely generated, by Corollary 10.29. Every finite set {x 1, x 2, …,x k} of generators is realized by disjoint n-surgeries

                   THE ODD-DIMENSIONAL SURGERY OBSTRUCTION
for i = 1, 2,…, k.

By the Poincaré Disc Theorem (9.14) it may be assumed that the target (2n+ 1)-dimensional geometric Poincare complex X is obtained from a (2n + 1)- dimensional geometric Poincaré pair (X 0, S 2n) by attaching a (2n + 1)-cell

X = X 0 D 2 n + 1 .
Any choice of generators x 1, x 2, …, xkKn(M) is realized by framed embeddings gi: Sn × D n+1M 2n+1 together with null-homotopies hi: fgi ≃ {*}: SnX.

Definition 12.6 A Heegaard splitting of an n-connected (2n + 1)-dimensional degree 1 normal map (f, b): M 2n+1X is an expression as a union

( f , b ) = ( f 0 , b 0 ) ( e , a ) : M 2 n + 1 = M 0 U X = X 0 D 2 n + 1 ,
with
U 2 n + 1 = # i = 1 k g i ( S n × D n + 1 ) M 2 n + 1 , M 0 = cl . ( M \ U ) , M 0 U = M 0 = U = # i = 1 k g i ( S n × S n ) ,
associated to a set of framed n-embeddings
g i : S n × D n + 1 M ( 1 i k )
with null-homotopies in X representing a set {x 1, x 2, …, xk} ⊂ Kn(M) of Z[π1(X)]-module generators. □ (p.308)
                   THE ODD-DIMENSIONAL SURGERY OBSTRUCTION

Every finite set of generators of K n(M) is realized by a Heegaard splitting.

Definition 12.7 The kernel formation of an n-connected (2n+1)-dimensional degree 1 normal map (f, b): M 2n+1X with respect to a Heegaard splitting is the (−1)n-quadratic formation over Z[π1(X)],

( K , λ , μ ; F , G ) = ( K n ( U ) , λ , μ ; K n + 1 ( U , U ) , K n + 1 ( M 0 , U ) ) ,
with ( K n ( U ) , λ , μ ) = H ( 1 ) n ( K n + 1 ( U , U ) ) the hyperbolic (–1)n-quadratic kernel form over Z[π1(X)] of the n-connected 2n-dimensional degree 1 normal map ∂US 2n. The lagrangians F, G are determined by the n-connected (2n + 1)-dimensional degree 1 normal maps of pairs
( e , a ) : ( U , U ) ( D 2 n + 1 , S 2 n ) , ( f 0 , b 0 ) : ( M 0 , U ) ( X 0 , S 2 n ) ,
with
F = im ( : K n + 1 ( U , U ) K n ( U ) ) [ π 1 ( X ) ] k , G = im ( : K n + 1 ( M 0 , U ) K n ( U ) ) [ π 1 ( X ) ] k .
 □

The Heegaard splittings and kernel formations of an n-connected (2n + 1)- dimensional degree 1 normal map (f, b): M 2n+1X are highly non-unique.

Proposition 12.8 For any kernel formation (K, λ, μ; F, G) of an n-connected (2n + 1)-dimensional degree 1 normal map (f, b): M 2n+1X there are natural identifications of Z[π1(X)]-modules:

k n ( M ) = K / ( F + G ) , K n + 1 ( M ) = F G .

Proof Immediate from the exact sequence

0 K n + 1 ( M ) K n + 1 ( M , U ) K n ( U ) K n ( M ) 0
and the identification of the map in the middle with the natural Z[π1(X)]-module morphism
K n + 1 ( M , U ) = K n + 1 ( M 0 , U ) = G K n ( U ) = K / F .
 □

(p.309) The kernel formations will be used in Section 12.4 to represent the surgery obstruction σ *(f, b) = (K, λ, μ; F, G) ∈ L 2n+1(Z[π1(X)]).

Example 12.9 The kernel formation of the 1-connected degree 1 normal map on the 3-dimensional lens space (8.39),

( f , b ) : L ( m , n ) 3 S 3 ,
is the (−1)-quadratic formation over Z given by
( H ( ) ; , im ( ( n m ) : * ) ) .
In particular,
  1. (i) L(1, 0) = S 3 with trivial formation ( H ( ) ; , ( 0 1 ) ) ,

  2. (ii) L(0, 1) = S 1 × S 2 with boundary formation ( H ( ) ; , ( 1 0 ) ) ,

  3. (iii) L(1, 2) = S 3 with trivial formation ( H ( ) ; , ( 2 1 ) ) ,

  4. (iv) L(2, 1) = R P 3 with boundary formation ( H ( ) ; , ( 1 2 ) ) . . □

Proposition 12.10 Let (f, b): M 2n+1X be an n-connected (2n+1)-dimensional normal map.

  1. (i) The kernel formations associated to all the Heegaard splittings of (f, b) are stably isomorphic.

  2. (ii) For n ⩾ 2 every formation in the stable isomorphism class is realised by a Heegaard splitting of (f, b).

  3. (iii) A kernel formation of (f, b) is trivial if (and for n ⩾ 2 only if) (f, b) is a homotopy equivalence. □

Proof (i) Consider first the effect on the kernel formation of changing the framed n-embeddings g i: S n × D n+1M 2n+1 representing a set {x 1, x 2, …, x k} of Z[π1(X)]-module generators for K n (M). Proposition 10.13 gives

K n ( M ) = π n + 1 ( f ) = I n + 1 ( f ) ,
so that the framed n-embeddings g i are unique up to regular homotopy. Any two sets of framed n-embeddings representing x i,
g i , g i : S n × D n + 1 · M 2 n + 1 ( 1 i k ) ,
are thus related by regular homotopies
d i : g i g i : S n × D n + 1 M 2 n + 1 ,
(p.310) with null-homotopies in X. Write the inclusions of the lagrangians associated to the two Heegaard splittings
( f , b ) = ( f 0 , b 0 ) ( e , a ) : M 2 n + 1 = M 0 U X , ( f , b ) = ( f 0 , b 0 ) ( e , a ) : M 2 n + 1 = M 0 U X
as
( γ δ ) : G = K n + 1 ( M 0 , U ) K n ( U ) = K n + 1 ( U , U ) K n ( U ) = F F * , ( γ δ ) : G = K n + 1 ( M 0 , U ) K n ( U ) = K n + 1 ( U , U ) K n ( U ) = F F * .
Let
α : F F , β : G G
be the Z[π1(X)]-module isomorphisms determined by the given basis elements, and let (F*, ψ) be a kernel split (−)n+1-quadratic form for the immersion
k S n × D n + 1 × I M × I ; ( y i , z i , t ) ( d i ( y i , z i , t ) , t )
(defined as in the proof of Proposition 11.42). The commutative diagram
                   THE ODD-DIMENSIONAL SURGERY OBSTRUCTION
defines an isomorphism of the kernel (−1)n-quadratic formations
( K n ( U ) , λ , μ ; K n + 1 ( U , U ) , K n + 1 ( M 0 , U ) ) = ( H ( 1 ) n ( F ) ; F , im ( ( γ δ ) : G F F * ) ) ( K n ( U ) , λ , μ ; K n + 1 ( U , U ) , K n + 1 ( M 0 , U ) ) = ( H ( 1 ) n ( F ) ; F , im ( ( γ δ ) : G F F * ) ) .

Next, consider the relationship between the kernel formations associated to two different sets of generators of K n(M). Proceed as in Wall [92,Chapter 6]. (p.311) Any two sets {x 1, x 2, …, x k}, {y 1, y 2, …, y l} of Z[π1(X)]-module generators for K n(M) are related by a sequence of elementary operations

{ x 1 , x 2 , , x k } { x 1 , x 2 , , x k , 0 } { x 1 , x 2 , , x k , y 1 } { x 1 , x 2 , , x k , y 1 , 0 } { x 1 , x 2 , , x k , y 1 , y 2 } { x 1 , x 2 , , x k , y 1 , y 2 , , y } { y 1 , y 2 , , y , x 1 , x 2 , , x k } { y 1 , y 2 , , y } ,
with y i = j a i j x j for some a ij ∈ Z[π1(X)]. Each of these operations has one of the following types:
  1. (1) adjoin (or delete) a zero,

  2. (2) permute the elements,

  3. (3) add to the last element a Z[π1(X)]-linear combination of the others.

The effect of (1) on the kernel formation of (f, b) is to add (or delete) a trivial formation
( H ( 1 ) n ( [ π 1 ( X ) ] ) ; [ π 1 ( X ) ] , [ π 1 ( X ) ] * ) ,
while (2) and (3) do not change it.

(ii) For n ⩾ 2 it is possible to realize every elementary operation geometrically.

(iii) A kernel formation for (f, b) is trivial if and only if K *(M) = 0. Now K *(M) = 0 if (and for n ⩾ 2 only if) (f, b) is a homotopy equivalence. □

The kernel formations of (f, b): M 2n+1X were obtained in 12.7 by working inside M, using Heegaard splittings. However, as described in Example 6.3 of Ranicki [75] (and pp. 71–72 of the second edition of Wall [92]) there is an alternative construction, working outside of M as follows.

Definition 12.11 A presentation of an n-connected (2n + 1)-dimensional degree 1 normal map (f, b): M 2n+1X is a degree 1 normal bordism

( ( e , a ) ; ( f , b ) , ( f , b ) ) : ( W 2 n + 2 ; M 2 n + 1 ) X × ( I ; { 0 } , { 1 } )
such that f′: M′ → X × {1} is n-connected and e: W → X × I is (n+1)-connected. □

The (2n + 2)-dimensional manifold with boundary W in a presentation has a handle decomposition on M of the type

W = M × I k ( n + 1 ) -handles D n + 1 × D n + 1 .
(p.312) The kernels K n+1(W), K n+1(W, M), K n+1(W, M′) are f.g. free Z[π1(X)]-modules of rank k. The cobordism (W; M, M′) is the trace of n-surgeries
g i : S n × D n + 1 M ( 1 i k ) ,
with null-homotopies in X representing a set {x 1, x 2, …, x k} of Z[π1(X)]-module generators of K n(M). The kernel modules K *(M), K *(M′) fit into exact sequences:
0 K n + 1 ( M ) K n + 1 ( W ) K n + 1 ( W , M ) K n ( M ) 0 0 K n + 1 ( M ) K n + 1 ( W ) K n + 1 ( W , M ) K n ( M ) 0.

Proposition 12.12 Let (f, b): M 2n+1X be an n-connected (2n + 1)-dimensional normal map.

  1. (i) There exist presentations (e, a): (W; M, M′) → X × (I; {0}, {1}).

  2. (ii) A presentation (e, a) determines a kernel (−1)n-quadratic formation for (f, b)

    ( H ( 1 ) n ( F ) ; F , G ) = ( H ( 1 ) n ( K n + 1 ( W , M ) ) ; K n + 1 ( W , M ) , K n + 1 ( W ) ) .
    The inclusion of the lagrangian,
    ( γ δ ) : G = K n + 1 ( W ) F F = K n + 1 ( W , M ) K n + 1 ( W , M ) ,
    has components
    γ : G = K n + 1 ( W ) inclusion * K n + 1 ( W , M ) = F , δ : G = K n + 1 ( W ) inclusion * K n + 1 ( W , M ) K n + 1 ( W , M ) K n + 1 ( W , M ) = F .

  3. (iii) For n ⩾ 2 every formation in the stable isomorphism class is realized by a presentation of (f, b).

Proof (i) A Heegaard splitting,

( f , b ) = ( f 0 , b 0 ) ( e , a ) : M 2 n + 1 = M 0 U X = X 0 D 2 n + 1 ,
(p.313) determines k n-surgeries U = # k S n × D n + 1 M on  ( f , b ) .
                   THE ODD-DIMENSIONAL SURGERY OBSTRUCTION
The trace of the surgeries is a presentation
( ( e , a ) ; ( f , b ) , ( f , b ) ) : ( W 2 n + 2 ; M 2 n + 1 , M 2 n + 1 ) X × ( I ; { 0 } , { 1 } )
The kernel formation of (f, b) given by 12.7 is
( H ( 1 ) n ( K n + 1 ( U , U ) ) ; K n + 1 ( U , U ) , K n + 1 ( M 0 , U ) ) = ( H ( 1 ) n ( F ) ; F , G ) .

(ii) Any presentation arises from a Heegaard splitting as in (i).

(iii) Combine (i), (ii) and 12.10. □

Note that turning a presentation of (f, b) around, and viewing it as a presentation of (f′, b′) gives the kernel formation ( H ( 1 ) n ( F ) ; F , G ) for (f′, b′) with

γ = ( 1 ) n + 1 δ : G = G F = F * , δ = γ : G = G f * = F .

Proposition 12.13 A kernel (− 1)n-quadratic formation of an n-connected (2n + 1)-dimensional degree 1 normal map (f, b): M 2n+1X is stably isomorphic to the boundary of a (−1)n+1-quadratic form if (and for n ⩾ 2 only if) (f, b) is bordant to a homotopy equivalence.

Proof Given a normal bordism

( ( e , a ) ; ( f , b ) , ( f , b ) ) : ( W 2 n + 1 ; M 2 n + 1 , M 2 n + + 1 ) X × ( I ; { 0 } , { 1 } ) ,
with (f′, b′): M′X a homotopy equivalence make (e, a) (n + 1)-connected by surgery below the middle dimension, with kernel (−1)n+1-quadratic form (p.314) (K n+1(W), λ w, μ w). This defines a presentation of (f, b) with
γ : K n + 1 ( W ) K n + 1 ( W , M )
an isomorphism which is used to identify
K n + 1 ( W ) = K n + 1 ( W , M ) = K n + 1 ( W , M ) = K n + 1 ( W , M ) ,
and
δ = λ W : K n + 1 ( W ) K n + 1 ( W , M ) = K n + 1 ( W ) ,
so that the kernel formation of (f, b) is the boundary
( H ( 1 ) n ( F ) ; F , G ) = ( K n + 1 ( W ) , λ W , μ W ) .

Conversely, suppose that n ⩾ 2 and that (f, b): M 2n+1X has a kernel formation which is stably isomorphic to the boundary of a (−1)n+1-quadratic form. By 12.12 it is possible to realize this boundary by a presentation

( ( e , a ) ; ( f , b ) , ( f , b ) ) : ( W 2 n + 2 ; M 2 n + 1 , M 2 n + 1 ) X × ( I ; { 0 } , { 1 } ) ,
with (f′, b′): M′X a homotopy equivalence. □

Example 12.14 An element

( δ ω , ω ) π n + 1 ( S O , S O ( n + 1 ) ) = π n + 2 ( B S O , B S O ( n + 1 ) ) = Q ( 1 ) n + 1 ( )
classifies an oriented (n + 1)-plane bundle ω: S n+1BSO(n+1) with a stable trivialization δω: ω ≃ {*}: S n+1BSO. As in 5.68 use (δω,ω) to define an n-surgery (g δω, gω) on the identity degree 1 normal map 1: S 2n+1S 2n+1, with trace
( ( e , a ) ; 1 , ( f , b ) ) : ( W 2 n + 2 ; S 2 n + 1 , S ( w ) 2 n + 1 ) S 2 n + 1 × ( I ; { 0 } , { 1 } ) .
The n-connected (2n + 1)-dimensional degree 1 normal map (f, b): S(ω) → S 2n+1 has kernel (−1)n-quadratic formation over Z
( H ( 1 ) n ( ) ; , im ( ( 1 χ ( ω ) ) : ) ) = ( , ( δ ω , ω ) ) ,
with χ(ω)=(1+(−1)(n+1)χ(δω). □

Proposition 12.15 Let

( ( e , a ) ; ( f , b ) , ( f , b ) ) : ( W 2 n + 2 ; M 2 n + 1 , M 2 n + 1 ) X × ( I ; { 0 } , { 1 } )
be an (n+l)-connected (2n+2)-dimensional normal bordism between n-connected (2n + 1)-dimensional degree 1 normal maps (f, b), (f′, b′). The kernel formations (K, λ, μ; F, G), (K′, λ′, μ′; F′, G′) of (f, b), (f′, b′) are related by a stable isomorphism
( K , λ , μ ; F , G ) ( K , λ , μ , F , G ) ( K n + 1 ( W ) , λ W , μ W ) .

(p.315) Proof The disjoint union

( f , b ) ( f , b ) : M M X X
is a degree 1 normal map, where – reverses orientations. The stable isomorphism of formations is determined by the degree 1 normal map of pairs
( ( e , a ) , ( f , b ) ( f , b ) ) : ( W 2 n + 2 , M M ) X × ( I , { 0 , 1 } ) ,
working as in the proof of 12.13. □

The kernel formation is also defined for an n-connected (2n+ 1)-dimensional degree 1 normal map

( f , b ) : ( M , M ) ( X , X )
which is a homotopy equivalence on the boundary, realizing a set {x 1, x 2, …, x k} of Z[π1(X)]-module generators of K n(M) as in 12.6 by a decomposition
( f , b ) = ( f 0 , b 0 ) ( e , a ) : ( M , M ) = ( M 0 ; M , U ) ( U , U ) ( X , X ) = ( X 0 , X , S 2 n ) ( D 2 n + 1 , S 2 n ) ,
with
U 2 n + 1 = # i = 1 k g i ( S n × D n + 1 ) M \ M , M 0 = cl . ( M \ U ) , M 0 U = U = # i = 1 k g i ( S n × S n ) .

There is also a version for normal bordisms, as follows.

Definition 12.16 Suppose given an n-connected (2n+1)-dimensional degree 1 normal bordism

( f , b ) : ( M 2 n + 1 ; N , N ) N × ( I ; { 0 } , { 1 } )
such that f| = 1: NN and f|: N′N is a homotopy equivalence. The Umkehr maps in this case are just
f ! = inclusion * : H * ( N ˜ ) H * ( M ˜ ) ,
the kernel Z[π1(N)]-modules are such that
K i ( M ) = H i ( M ˜ , N ˜ ) = 0 for i n , n + 1 ,
and the cobordism (M; N, N′) has an (n, n + 1)-index handle decomposition (8.23)
M = N × I k D n × D n + 1 k D n + 1 × D n .
(p.316)
  1. (i) A Heegaard splitting for (f, b) is an expression as a union

    ( f , b ) = ( e , a ) ( f 0 , b 0 ) : ( M ; N , N ) = ( U ; N , + U ) ( M 0 , + U , N ) ( N × [ 0 , 1 / 2 ] ; { 0 } , { 1 / 2 } ) ( N × [ 1 / 2 , 1 ] ; { 1 / 2 } , { 1 } )
    determined by a choice of handle decomposition as above, with
    U 2 n + 1 = N × I k D n × D n + 1 M 0 U = + U = N # # k ( S n × S n ) , M 0 = + U × I U k D n + 1 × D n .
    The ith handle represents x iK n(M) = H n(M̃, Ñ), with {x 1, x 2,…, x k} ⊂ K n(M) a set of Z[π1(N)]-module generators.
                       THE ODD-DIMENSIONAL SURGERY OBSTRUCTION

  2. (ii) The kernel formation associated to a Heegaard splitting as in (i) is

    ( K , λ , μ ; F , G ) = ( K n ( U ) , λ , μ ; K n + 1 ( U , U ) , K n + 1 ( M 0 , U ) )
    with ( K n ( U ) , λ , μ ) = H ( 1 ) n ( K n + 1 ( U , U ) ) .  □

As in the closed case (12.6) every finite set of generators of K n(M) is realized by a Heegaard splitting of (f, b), and so determines a kernel formation (K, λ, μ; F, G).

Proposition 12.17 (Realization of formations) Let N 2n be a 2n-dimensional manifold with fundamental group π 1(N) = π, with n ⩾ 2. Every (−1)n-quadratic formation (K, λ, μ; F, G) over Z[π] is realized as a kernel formation (12.16) of an n-connected (2n + 1)-dimensional degree 1 normal bordism

( f , b ) : ( M 2 n + 1 ; N 2 n , N 2 n ) N × ( I ; { 0 } , { 1 } )
with (f, b)| = 1: NN and (f, b)|: N′N a homotopy equivalence.

(p.317) Proof By 11.51 the form (K, λ, μ) is isomorphic to the hyperbolic form H ( 1 ) n ( F ) , so there is no loss of generality in taking ( K , λ , μ ) = H ( 1 ) n ( F ) . . Let k ⩾ 0 be the rank of the f.g. free Z[π]-modules F, G, so that

F G [ π ] k .
Let
( f , b ) : ( U 2 n + 1 ; N , + U ) N × ( [ 0 , 1 / 2 ] ; { 0 } , { 1 / 2 } )
be the (2n + 1)-dimensional normal bordism defined by the trace of k trivial (n − 1)-surgeries on (f, b), with
U = N × I k n -handles D n × D n + 1 , ( f , b ) | = 1 # std . : + U = N # # k S n × S n N .
Realize the lagrangian of the kernel form
G ( K n ( + U ) λ , μ ) = H ( 1 ) n ( F )
by a (2n + 1)-dimensional normal bordism
( f , b ) : ( M 0 2 n + 1 ; + U , N ) N × ( [ 1 / 2 , 1 ] ; { 1 / 2 } , { 1 } )
defined by the trace of k n-surgeries on (f′, b′)|, with
M 0 = + U × I k ( n + 1 ) -handles D n + 1 × D n , G = im ( K n + 1 ( M 0 , + U ) K n ( + U ) ) K n ( + U ) .
The effect is a homotopy equivalence (f′, b′)|: N′N, since the kernel form is
( K n ( N ) , λ , μ ) = ( G / G , [ λ ] , [ μ ] ) = 0.
The n-connected (2n + 1)-dimensional degree 1 normal map of pairs
( f , b ) = ( f , b ) ( f , b ) : ( M 2 n + 1 ; N , N ) = ( U ; N , + U ) ( M 0 ; + U , N ) N × ( I ; { 0 } , { 1 } )
realizes the formation (K, λ, μ; F, G). □

12.3 The odd-dimensional L-groups

The odd-dimensional surgery obstruction groups L 2*+1 (A) are now defined using formations. The surgery obstruction of an odd-dimensional normal map will be defined in Section 12.4 using the kernel formation of Section 12.2, and it will be proved that for n ⩾ 2 an n-connected (2n + 1)-dimensional degree 1 normal map (p.318) (f, b): M 2n+1X is bordant to a homotopy equivalence if and only if the stable isomorphism class of kernel (−1)n-quadratic formations contains the ‘boundary’ of a (−1)n+1-quadratic form (= the kernel form of the (2n + 2)-dimensional trace), in the following sense:

Definition 12.18 Let (K, λ, μ) be a (−ε)-quadratic form.

  1. (i) The graph lagrangian of (K, λ, μ) is the lagrangian

    Γ ( K , λ ) = { ( x , λ ( x ) ) K K | x K }
    in the hyperbolic ε-quadratic form H (K)

  2. (ii) The boundary of (K, λ, μ) is the graph ε-quadratic formation

    ( K , λ , μ ) = ( H ( K ) ; K , Γ ( K , λ ) ) .
     □

The graph lagrangian Γ(K, λ) and the boundary formation (K, λ, μ) depend only on the even ε-symmetric form (K, λ), and not on the ε-quadratic function μ. Note that the form (K, λ, μ) may be singular, that is the A-module morphism λ: KK * need not be an isomorphism.

Proposition 12.19

  1. (i) The graphs Γ(K, λ) of (−ε)-quadratic forms (K, λ, μ) are precisely the lagrangians of H ε(K) which are the direct complements of K *.

  2. (ii) An ε-quadratic formation (K, λ, μ; F, G) is isomorphic to a boundary if and only if (K, λ, μ) has a lagrangian H which is a direct complement of both the lagrangians F, G.

Proof (i) The direct complements of K * in KK * are the graphs

L = { ( x , λ ( x ) ) K K | x K }
of A-module morphisms λ: KK *, with
L = { ( y , λ ( y ) ) K K | y K } .
Thus L = L if and only if λ = −∈λ *, with μ H ( K ) ( L ) = 0 if and only if λ admits a (−∈)-quadratic refinement μ.

(ii) For the boundary (F, φ, Θ) of a (−ε)-quadratic form (F, φ, Θ) the lagrangian F * of H ε(F) is a direct complement of both the lagrangians F, Γ(F, φ). Conversely, suppose that (K, λ, μ; F, G) is such that there exists a lagrangian H in (K, λ, μ) which is a direct complement to both F and G. By the proof of Proposition 12.5 (i) there exists an isomorphism of formations

f : ( H ( F ) ; F , F ) ( K , λ , μ ; F , H ) ,
which is the identity on F. Now f −1 (G) is a lagrangian of H ε(F) which is a direct complement of F *, so that by (i) it is the graph Γ(F, φ) of a (−ε)-quadratic (p.319) form (F, Φ, Θ), and f defines an isomorphism of ε-quadratic formations
f : ( F , φ , θ ) = ( H ( F ) ; F , Γ ( F , φ ) ) ( K , λ , μ ; F , G ) .
 □

Definition 12.20 The ε-quadratic formations (K, λ, μ; F, G), (K′, λ′, μ′; F′, G′) over A are cobordant,

( K , λ , μ ; F , G ) ˜ ( K , λ , μ ; F , G ) .
if there exists a stable isomorphism
[ f ] : ( K , λ , μ ; F , G ) B ( K , λ , μ ; F , G ) B ,
with B, B′ boundaries. □

Proposition 12.21

  1. (i) Cobordism is an equivalence relation on ε-quadratic formations over A.

  2. (ii) For any lagrangians F, G, H in a nonsingular ε-quadratic form (K, λ, μ)

    ( K , λ , μ ; F , G ) ( K , λ , μ ; G , H ) ˜ ( K , λ , μ ; F , H ) .

  3. (iii) For any ε-quadratic formation (K, λ, μ; F, G)

    ( K , λ , μ ; F , G ) ( K , λ , μ ; G , F ) ˜ 0 , ( K , λ , μ ; F , G ) ( K , λ , μ ; F , G ) ˜ 0.

Proof (i) Clear.

(ii) (Taken from Proposition 9.14 of [69]). Choose lagrangians F *, G *, H * in (K, λ, μ) complementary to F, G, H, respectively. The ε-quadratic formations (K i, λ i, μ i; F i, G i) (1 ≤ i ≤ 4) defined by

( K 1 , λ 1 , μ 1 ; F 1 , G 1 ) = ( K , λ , μ ; G , G ) , ( K 2 , λ 2 , μ 2 ; F 2 , G 2 ) = ( K K , λ λ , μ μ ; F F , H G ) ( K K , λ λ , μ μ ; Δ K , H G ) , ( K 3 , λ 3 , μ 3 ; F 3 , G 3 ) = ( K K , λ λ , μ μ ; F F , G G ) , ( K 4 , λ 4 , μ 4 ; F 4 , G 4 ) = ( K K , λ λ , μ μ ; G G , H G ) , ( K K , λ λ , μ μ ; Δ K , H G )
are such that
( K , λ , μ ; F , G ) ( K , λ , μ ; G , H ) ( K 1 , λ 1 , μ 1 ; F 1 , G 1 ) ( K 2 , λ 2 , μ 2 ; F 2 , G 2 ) = ( K , λ , μ ; F , H ) ( K 3 , λ 3 , μ 3 ; F 3 , G 3 ) ( K 4 , λ 4 , μ 4 ; F 4 , G 4 ) .
Each of (K i, λ i, μ i; F i, G i) (1 ≤ i ≤ 4) is isomorphic to a boundary, since there exists a lagrangian H i in (K i, λ i, μ i) complementary to both F i and G i, so that (p.320) 12.19 (ii) applies and (K i, λ i, μ i; F i, G i) ˜ 0. Explicitly, take
H 1 = G K 1 = K , H 2 = Δ K K K 2 = ( K K ) ( K K ) , H 3 = Δ K K 3 = K K , H 4 = Δ K K K 4 = ( K K ) ( K K ) .

(iii) By (ii)

( K , λ , μ ; F , G ) ( K , λ , μ ; G , F ) ˜ ( K , λ , μ ; F , F ) = ( F , 0 , 0 ) ˜ 0 ,
and by Proposition 12.5 (iii)
( K , λ , μ ; G , F ) ˜ ( K , λ , μ ; F , G ) .
 □

Remark 12.22 The identity of 12.21 (ii)

( K , λ , μ ; F , G ) ( K , λ , μ ; G , H ) ˜ ( K , λ , μ ; F , H )
is the L-theoretic analogue of the Whitehead Lemma 8.2. See Lemma 6.2 of Wall [92] and the commentary on pp. 72–73 of [92] for the geometric motivation. □

Definition 12.23 The (2n + 1)-dimensional L-group L 2n+1(A) of a ring with involution A is the group of cobordism classes of (−1)n-quadratic formations (K, λ, μ; F, G) over A, with addition and inverses given by

( K 1 , λ 1 , μ 1 ; F 1 , G 1 ) + ( K 2 , λ 2 , μ 2 ; F 2 , G 2 ) = ( K 1 K 2 , λ 1 λ 2 , μ 1 μ 2 ; F 1 F 2 , G 1 G 2 ) , ( K , λ , μ ; F , G ) = ( K , λ , μ ; F , G ) L 2 n + 1 ( A ) .
 □

Since L 2n+1(A) depends on the residue n(mod 2), only two L-groups have actually been defined, L 1(A) and L 3(A).

Example 12.24 Kervaire and Milnor [38] proved that the odd-dimensional L-groups of Z are trivial

L 2 n + 1 ( ) = 0.
See Example 12.44 below for an outline of the computation. □

Remark 12.25 Chapter 22 of Ranicki [71] is an introduction to the computation of the odd-dimensional surgery obstruction groups of finite groups π, with

L 2 n + 1 ( [ π ] ) = ( 2 primary torsion ) .
See Hambleton and Taylor [30] for a considerably more complete account. □

(p.321) Example 12.26 The odd-dimensional L-groups of Z[Z 2] with the oriented involution = T are given by

L 2 n + 1 ( [ 2 ] ) = { 0 , if n 0 ( mod 2 ) , 2 , if n 1 ( mod 2 ) .
 □

12.4 The odd-dimensional surgery obstruction

It was shown in Section 10.4 that every (2n+1)-dimensional degree 1 normal map is bordant to an n-connected degree 1 normal map. As in the even-dimensional case considered in Section 11.5 there is an obstruction to the existence of a further bordism to an (n + 1)-connected degree 1 normal map (= homotopy equivalence), which is defined as follows.

Definition 12.27 The surgery obstruction of an n-connected (2n + 1)-dimensional degree 1 normal map (f, b): M 2n+1X is the cobordism class of a kernel (−1)n-quadratic formation over Z[π1(X)]:

σ * ( f , b ) = ( K , λ , μ ; F , G ) L 2 n + 1 ( [ π 1 ( X ) ] ) .
 □

The main result of this section is that σ *(f, b) = 0 if (and for n ⩾ 2 only if) (f, b) is bordant to a homotopy equivalence. It is clear that if (f, b) is a homotopy equivalence then σ * (f, b) = 0, for then (K, λ, μ; F, G) is a trivial formation.

Proposition 12.28 The surgery obstructions of bordant n-connected (2n + 1)-dimensional degree 1 normal maps (f, b): M 2n+1X, (f′, b′): M′ 2n+1X are the same:

σ * ( f , b ) = σ * ( f , b ) L 2 n + 1 ( [ π 1 ( X ) ] ) .

Proof By 12.15 the kernel formations (K, λ μ; F, G), (K′, λ′, μ′; F′, G′) of (f, b), (f′, b′) are related by a stable isomorphism

( K , λ , μ ; F , G ) ( K , λ , μ , F , G ) ( K n + 1 ( W ) , λ W , μ W )
with (K n+1(W), λ W, μ W) the kernel (−1)n+1-quadratic form of an (n + 1)-connected normal bordism
( ( e , a ) ; ( f , b ) , ( f , b ) ) : ( W 2 n + 2 , M 2 n + 1 , M 2 n + 1 ) X × ( I ; { 0 } , { 1 } ) . n
By Proposition 12.21 (iii)
( K , λ , μ ; F , G ) ( K , λ , μ ; F , G ) = 0 L 2 n + 1 ( [ π 1 ( X ) ] ) .
(This can also be proved geometrically, by considering the (n + 1)-connected normal bordism (e, a) obtained from
( f , b ) × 1 : M × ( I ; { 0 } , { 1 } ) X × ( I ; { 0 } , { 1 } )
(p.322) by n-surgeries on the interior killing K n(M × I) = K n(M), with a stable isomorphism
( K , λ , μ ; F , G ) ( K , λ , μ ; F , G ) ( K n + 1 ( W ) , λ W , μ W )
as above.) The surgery obstructions are such that
σ * ( f , b ) = ( K , λ , μ ; F , G ) = ( K , λ , μ ; F , G ) = ( K , λ , μ ; F , G ) = σ * ( f , b ) L 2 n + 1 ( [ π 1 ( X ) ] ) .
 □

Theorem 12.29 A (2n + 1)-dimensional degree 1 normal map of pairs

( f , b ) : ( M 2 n + 1 , M ) ( X , X )
with ∂f: ∂M∂X a homotopy equivalence has a rel ∂ surgery obstruction
σ * ( f , b ) L 2 n + 1 ( [ π 1 ( X ) ] )
such that σ *(f, b) = 0 if (and for n ⩾ 2 only if) (f, b) is bordant rel ∂ to a homotopy equivalence of pairs.

Proof The surgery obstruction of (f, b) is defined by

σ * ( f , b ) = ( K , λ , μ , F , G ) L 2 n + 1 ( [ π 1 ( X ) ] ) ,
with (K′, λ′, μ′; F′, G′) a kernel (−1)n-quadratic formation for any n-connected degree 1 normal map (f′, b′): (M′, ∂M) → (X, ∂X) bordant to (f, b) relative to the boundary, with ∂f′ = ∂f, exactly as in the closed case ∂M = ∂ X = ∅ in 12.27. The rel version of 12.28 shows that the surgery obstruction is a normal bordism invariant, which is 0 for a homotopy equivalence. Conversely, assume that n ⩾ 2 and σ *(f, b) = 0 ∈ L 2n+1(Z[π1(X)]), so that (f, b) has a kernel (−1)n-quadratic formation (K, λ, μ; F, G) with a stable isomorphism,
( K , λ , μ ; F , G ) B B ,
for some boundary formations B = (H, Φ, Θ), B′ = (H′, Φ′, Θ′), with H, H′ f.g. free Z[π1(X)]-modules of ranks k, k′ (say). As in the proof of Proposition 11.42 use the (−1)n+1-quadratic form (H, Φ, Θ) to perform k n-surgeries on (f, b): MX killing 0 ∈ K n(M), such that the trace,
( ( e , a ) ; ( f , b ) , ( f , b ) ) : ( W 2 n + 2 ; M 2 n + 1 , M 2 n + 1 ) X × ( I ; { 0 } , { 1 } ) ,
is (n + 1)-connected with kernel (−1)n+1-quadratic form
( K n + 1 ( W ) , λ W , μ W ) = ( H , φ , θ ) .
The effect is an n-connected degree 1 normal map (f′, b′): M′X with kernel formation (K, λ, μ; F, G) ⊕ B stably isomorphic to the boundary B′. By Proposition 12.13 K n(M′) can be killed by k′ n-surgeries on (f′, b′), so that (f, b) is bordant to a homotopy equivalence. □

(p.323) Corollary 12.30 Let π be a finitely presented group with an orientation character w: π → Z2, and let n ⩾ 2. Every element xL 2n+1(Z[π]) is the rel ∂ surgery obstruction x = σ *(f, b) of an n-connected (2n + 1)-dimensional degree 1 normal bordism (f, b): M 2n+1X with1(X), w(X)) = (π, w).

Proof By Proposition 11.75 there exists a closed 2n-dimensional manifold N with (π1(N), w(N)) = (π, w). By Proposition 12.17 every (−1)n-quadratic formation (K, λ, μ; F, G) representing x is realized as the kernel formation of an n-connected (2n + 1)-dimensional degree 1 normal bordism,

( f , b ) : ( M 2 n + 1 ; N 2 n , N 2 n ) N × ( I ; { 0 } , { 1 } ) ,
with (f, b)| = 1: N → N and (f, b)|: N′ → N a homotopy equivalence. The rel surgery obstruction is
σ * ( f , b ) = ( K , λ , μ ; F , G ) = x L 2 n + 1 ( [ π ] ) .
 □

Remark 12.31 An ε-quadratic formation over a ring with involution A is null-cobordant if and only if it is stably isomorphic to the boundary of a (−∈)- quadratic form (Corollary 9.12 of Ranicki [75]). A (−l)n-quadratic formation (K, λ, μ; F, G) is, thus, such that

( K , λ , μ l F , G ) = 0 L 2 n + 1 ( A )
if and only if (K, λ, μ; F, G) is stably isomorphic to the boundary (H, Φ, Θ) of a (−1)n+1-quadratic form (H, Φ, Θ). For a group ring A = Z[π] this can be proved geometrically, using Theorem 12.29 and Corollary 12.30. □

12.5 Surgery on formations

The odd-dimensional surgery obstruction theory developed in Section 12.4 is somewhat indirect—it is hard to follow through the algebraic effect of geometric surgeries. This will now be made easier, using algebraic surgery on formations.

If (f, b): M → X, (f′, b′): M′ → X are n-connected (2n + 1)-dimensional degree 1 normal maps such that (f′, b′) is obtained from (f, b) by an n-surgery then a kernel (−l)n-quadratic formation for (f′, b′) can be obtained by an algebraic surgery on a kernel (−1)n-quadratic formation for (f, b). The geometric surgeries on (f, b) correspond to algebraic surgeries on a kernel formation, as in the even-dimensional case considered in Section 11.3. However, odd-dimensional surgery behaves somewhat differently from even-dimensional surgery. In both cases, the aim of performing surgery is to make the kernel modules as small as possible. Given an ε-quadratic form (K, λ, μ) over A it is possible to kill an element xK if and only if μ(x) = 0, with unique effect: if x ≠ 0 generates a (p.324) direct summand <x> ⊂ K the effect of the surgery is a cobordant form (K′, λ′, μ′) with

( K , λ , μ ) = ( < x > / < x > , [ λ ] , [ μ ] ) , ( K , λ , μ ) ( K , λ , μ ) H ( A ) , rank A ( K ) = rank A ( K ) 2 < rank A ( K ) .

Given an ε-quadratic formation (K, λ, μ; F, G) it is possible to kill every element in the kernel module xK/(F + G) by algebraic surgery, but there are many choices in carrying out such a surgery, and the effect of any such surgery may result in a formation (K′, λ′, μ′; F′, G′) with kernel module K′/(F′ + G′) bigger than K/(F+G). In the context of geometric surgery consider the trace of k n-surgeries on an n-connected (2n+1)-dimensional normal map (f, b): MX killing x 1, x 2,…,x kK n(M):

( ( F , B ) ; ( f , b ) , ( f , b ) ) : ( W 2 n + 2 ; M 2 n + 1 , M 2 n + 1 ) X × ( I ; { 0 } , { 1 } )
and let (K,λ, μ; F, G), (K′, λ′, μ′; F′, G′) be kernel (−1)n-quadratic formations for (f, b), (f′, b′). Proposition 10.25 (iii) gives a commutative braid of exact sequences
                   THE ODD-DIMENSIONAL SURGERY OBSTRUCTION
and a set of Z[π1(X)]-module generators {x′ 1, x′ 2,…,x′ k} ⊂ ker(K n(M′) → K n(W)) with
K n + 1 ( M ) = F G , K n ( M ) = K / ( F + G ) , K n + 1 ( M ) = F G , K n ( M ) = K / ( F + G ) , K n ( W ) = K n ( M ) / < x 1 , x 2 , , x x > = K n ( M ) / < x 1 , x 2 , , x x > .
The different effects of killing x 1, x 2,…,x k correspond to the different ways of framing n-embeddings g i: S nM 2n+1 representing x i, or equivalently to (p.325) the different extensions of g i to framed n-embeddings i: S n × D n+1M 2n+1. Every set of Z[π1(X)]-module generators {x 1,x 2,…,x k} ⊂ K n(M) can be killed by n-surgeries with (n + 1)-connected trace (i.e. K n(W) = 0) but in general the effect (f′, b′): M′ → X will not be a homotopy equivalence, with K n(M′) ≠ 0.

In order to keep track of algebraic surgeries on formations it is convenient to work with the following refinement of the notion of a formation.

Definition 12.32 (i) A split ε-quadratic formation over A

( F , G ) = ( F , ( ( γ δ ) , θ ) G )
is given by f.g. free A-modules F, G, morphisms γ: GF, δ: GF * and a (−∈)-quadratic form (G, Θ) such that
  1. (a) γ * δ = Θ − ∈Θ*: GG *,

  2. (b) the sequence

    0 G ( γ δ ) F F * ( δ * γ * ) G * 0
    is exact.

Equivalently,
( ( γ δ ) , θ ) : ( G , 0 ) H ( F )
is a morphism of split ε-quadratic forms which is the inclusion of a lagrangian.

(ii) An isomorphism of split ε-quadratic formations over A

( α , β , χ ) : ( F , G ) ( F . G )
is given by isomorphisms α: FF′, β: GG′ and a (−∈)-quadratic form (F *, χ) such that the diagram
                   THE ODD-DIMENSIONAL SURGERY OBSTRUCTION
commutes. Thus,
f = ( α α ( χ χ * ) 0 ( α * ) 1 ) : H ( F ) H ( F )
is an isomorphism of hyperbolic ε-quadratic forms with f(F) = F′, f(G) = G′.

(p.326) (iii) A split ε-quadratic formation (F, G) is trivial if it is isomorphic to

( F , F * ) = ( F , ( ( 0 1 ) , 0 ) F * ) .

(iv) A stable isomorphism of split ε-quadratic formations over A

[ α , β , χ ] : ( F , G ) ( F , G )
is an isomorphism of the type
( α , β , χ ) : ( F , G ) ( H , H ) ( F , G ) ( H , H ) ,
with (H, H *), (H′, H′ *) trivial split formations.

(v) The boundary of a split (−∈)-quadratic form (K, ψ) is the graph split ε-quadratic formation

( K , ψ ) = ( K , ( ( 1 ψ ψ ) , ψ ) K ) .

(vi) Split ε-quadratic formations (F,G), (F′,G′) are cobordant if there exists a stable isomorphism

[ α , β , χ ] : ( F , G ) ( K , ψ ) ( F . G ) ( K , ψ )
for some split (−∈)-quadratic forms (K, ψ), (K′,ψ′). □

Proposition 12.33

  1. (i) A split ε-quadratic formation (F, G) is isomorphic to a boundary if and only if there exists a split (−∈)-quadratic form (F *, χ) such that the morphism γ + (χ − εχ*)δ: GF is an isomorphism.

  2. (ii) A split formation (F, G) is stably isomorphic to 0 if and only if δ: G → F* is an isomorphism.

  3. (iii) Cobordism is an equivalence relation on split ε-quadratic formations over A.

  4. (iv) The cobordism group of split (−1)n-quadratic formations over A is isomorphic to L 2n+1(A).

  5. (v) For any split (−1)n-quadratic formation ( F , G ) = ( F , ( ( γ δ ) , θ ) G ) over A

    ( F , G ) = ( F , ( ( γ δ ) , θ ) G ) , ( F , G ) = ( F * , ( ( δ ( 1 n γ ) ) , θ ) G ) L 2 n + 1 ( A ) .

(p.327) Proof (i) If (α, β, χ): (F, G) → ∂(K,ψ) is an isomorphism of split formations then

α 1 β = γ + ( χ χ * ) δ : G F
is an A-module isomorphism.

For the converse, consider first the special case when γ is an isomorphism. There is defined an isomorphism of split formations

( γ , 1 , χ ) : ( G , θ ) ( F , G ) .
More generally, if γ′ = γ+(χ−∈χ *)δ: GF is an isomorphism there is defined an isomorphism of split formations
( 1 , 1 , χ ) : ( F , G ) ( F , G ) = ( F , ( ( γ δ ) , θ + δ * χ δ ) G )
and (F′, G′) is isomorphic to a boundary by the special case.

(ii) If (α, β, X): (F, G) → (K, K *) is an isomorphism of split formations then

δ = α * β : G F *
is an A-module isomorphism.

Conversely, if δ is an isomorphism there is defined an isomorphism of split formations,

( 1 , δ , ( δ * ) 1 θ δ 1 ) : ( F , G ) ( F , F * ) .

(iii) Clear.

(iv) A split ε-quadratic formation (F, G) determines an ε-quadratic formation

( H ( F ) ; F , im ( ( γ δ ) : G F F * ) ) ,
and an isomorphism of split formations determines an isomorphism of formations. Every ε-quadratic formation (K, λ, μ; F, G) is isomorphic to one of this type, by 12.3, and a (stable) isomorphism of formations
f : ( K , λ , μ ; F , G ) ( K , λ , μ ; F , G )
lifts to a (stable) isomorphism of split formations
( α , β , γ ) : ( F , G ) ( F , G ) .
So the only essential difference between a formation and a split formation is the choice of ‘Hessian’ form Θ. Suppose, given split ε-quadratic formations
Φ = ( F , ( ( γ δ ) , θ ) G ) , Φ = ( F , ( ( γ δ ) , θ ) G )
(p.328) with different Θ, Θ′ such that
γ * δ = θ θ * = θ θ : G G * .
Let
Φ ˜ = ( F * , ( ( γ ˜ δ ˜ ) , θ ˜ ) G * )
be the split ε-quadratic formation given by an extension (provided by 11.51) of the inclusion of the lagrangian,
( ( γ δ ) , θ ) : ( G , θ ) H ( F ) = ( F F * , ( 0 1 0 0 ) ) ,
to an isomorphism of hyperbolic split ε-quadratic forms
( ( γ δ ˜ δ γ ˜ ) , ( θ 0 δ ˜ * δ θ ˜ ) ) : H ( G ) = ( G G * , ( 0 1 0 0 ) ) H ( F ) = ( F F * , ( 0 1 0 0 ) ) .
The split ε-quadratic formation
Φ Φ ˜ = ( F F * , ( ( ( γ 0 0 γ ˜ ) ( δ 0 0 δ ˜ ) ) , ( θ 0 0 θ ˜ ) ) G G * )
is isomorphic to a boundary by (i), since
( γ 0 0 γ ˜ ) + ( 0 1 0 ) ( δ 0 0 δ ˜ ) = ( γ δ ˜ δ γ ˜ ) : G G * F F *
is an isomorphism. Similarly for the split formation Φ′ ⊕ Φ˜. The split formation Φ ⊕ Φ′ ⊕ Φ˜ is cobordant to both Φ and Φ′, which are thus cobordant to each other.

(v) These identities follow from 12.21 (ii)+(iii). (Alternatively, note that there exist stable isomorphisms

( F , ( ( γ δ ) , θ ) G ) ( F , ( ( γ δ ) , θ ) G ) ( G F , ( θ 0 δ 0 ) ) ( F , ( ( γ δ ) , θ ) G ) ( F * , ( ( δ ( 1 ) n + 1 γ ) , θ ) G ) ( G , θ ) .
The construction of such stable isomorphisms are exercises for the reader.) □

Definition 12.34 The data (H, X, j) for an algebraic surgery on a split ε-quadratic formation (F, G) is a split (−∈)-quadratic form (H, X) together with a (p.329) morphism j: F → H *. The effect of the algebraic surgery is the split ε-quadratic formation (F′, G′) with

γ = ( γ 0 0 1 ) : G = G H F = F H , δ = ( δ j j γ χ χ ) : G = G H F = F H , θ = ( θ 0 j γ χ ) : G = G H G * = G * H * .
 □

Proposition 12.35 (i) If (F 1, G 1), (F 2, G 2), (F 3, G 3) are split ε-quadratic formations such that (F i+1, G i+1) is stably isomorphic to the effect of an algebraic surgery on (F i, G i) (i = 1, 2) then (F 3, G 3) is stably isomorphic to the effect of an algebraic surgery on (F 1, G 1).

(ii) Split ε-quadratic formations (F, G), (F′, G′) are cobordant if and only if (F′, G′) is stably isomorphic to the effect of an algebraic surgery on (F, G).

(iii) A split (−1)n-quadratic formation (F, G) is such that (F, G) = 0 ∈ L 2n+1(A) if and only if there exist algebraic surgery data (H, χ, j) such that

δ = ( δ ( 1 ) n + 1 j * j γ χ + ( 1 ) n + 1 χ * ) : G = G H F * = F * H *
is an isomorphism, in which case (F, G) is stably isomorphic to the boundary ( G H , ( θ 0 j γ χ ) ) .

Proof (i) Exercise for the reader!

(ii) Suppose first that (F, G), (F′, G′) are cobordant, so that there exists a stable isomorphism

( F , G ) ( H , χ ) ( F , G ) ( H , χ )
for some (—∈)-quadratic forms (H, χ), (H′, χ′). Now (F, G)⊕∂(H, χ) is the effect of the algebraic surgery on (F, G) with data (H, χ, 0), and (F′, G′) is stably isomorphic to the effect of the algebraic surgery on (F′, G′) ⊕ ∂ (H′, χ′) with data (H′, χ′, j′ = (0 1): F′H′ *H′ *). It now follows from (i) that (F′, G′) is stably isomorphic to the effect of an algebraic surgery on (F, G).

Conversely, suppose that (F′, G′) is the effect of an algebraic surgery on (F, G) with data (H, χ, j). By 12.33 (v) (F′, G′) is cobordant to the split formation

( F * , G ) = ( F * , ( ( δ γ ) , θ ) G ) .
Now (F′ *, G′) is isomorphic to (F *, G) ⊕ (H *, H), and (F *, G) is cobordant to (F, G) (by 12.33 (v)), so that (F′, G′) is cobordant to (F, G).

(p.330) (iii) By (ii) a split (−1)n-quadratic formation (F, G) is null-cobordant if and only if there exists data (H, χ, j) such that the effect of the algebraic surgery (F′, G′) is trivial. □

All this can now be applied to surgery on highly-connected odd-dimensional normal maps.

Proposition 12.36 Let (f, b): M 2n+1X be an n-connected (2n+1)-dimensional degree 1 normal map. (i) A presentation (12.11) of (f, b),

( { e , a } ; ( f , b ) , ( f ^ , b ^ ) ) : ( W 2 n + 2 ; M 2 n + 1 , M ^ 2 n + 1 ) X × ( I ; { 0 } , { 1 } ) ,
determines a kernel split (−1)n -quadratic formation over Z[π1(X)]
( F , ( ( γ δ ) , θ ) G ) = ( K n + 1 ( W , M ) , K n + 1 ( W ) ) ,
with
γ = inclusion * : G = K n + 1 ( W ) F = K n + 1 ( W , M ^ ) , δ = inclusion * : G = K n + 1 ( W ) F * = K n + 1 ( W , M ) , θ = μ W : G = K n + 1 ( W ) G * = K n + 1 ( W , W ) , γ * δ = θ + ( 1 ) n + 1 θ * = λ W : G = K n + 1 ( W ) G * = K n + 1 ( W , W ) ,
and exact sequences
0 K n + 1 ( M ^ ) G γ F K n ( M ^ ) 0 , 0 K n + 1 ( M ) G δ F * K n ( M ) 0.

(ii) The surgery obstruction of (f, b) is the cobordism class

σ * ( f , b ) = ( F , G ) L 2 n + 1 ( [ π 1 ( X ) ] )
of the kernel split formation (F, G) constructed in (i) from any presentation of (f, b).

(iii) The effect of l simultaneous geometric n-surgeries on (f, b) killing x 1, x 2, …, x lK n(M) is a bordant n-connected (2n + 1)-dimensional normal map (f′, b′): M′ → X with kernel split formation (F′, G′) obtained by algebraic surgery on (F, G) with data (H, χ, j) such that

[ j * ] = ( x 1 , x 2 x ) : H = [ π 1 ( X ) ] K n ( M ) = coker ( δ : G F * ) .

(iv) For n ⩾ 2 algebraic surgeries on (F, G) are realized by geometric surgeries on (f, b).

(p.331) (v) Let x 1, x 2, …, x 1K n+1(M) be as in (iii) (or (iv)). If there exist y 1, y 2, …, y lK n+1(M) such that

λ ( x i , y i ) = 1 [ π 1 ( X ) ] ( 1 i ) ,
with λ: K n(M) × K n+1(M) → Z[π1(X)] the homology intersection pairing (10.22) then
K n ( M ) = K n ( M ) / < x 1 , x 2 , , x > .

Proof (i) The split formation (F, G) is just the split version of the kernel formation 12.12 (ii).

(ii) Immediate from (i) and 12.29.

(iii) Let (f′, b′): M′ 2n+1X be the effect of l n-surgeries on (f, b) killing x 1, x 2, …, x lK n(M). The trace degree 1 normal bordism,

( ( g , c ) ; ( f , b ) , ( f , b ) ) : ( N 2 n + 2 ; M , M ) X × ( I ; { 0 } , { 1 } ) ,
is n-connected, and such that
( f , b ) | = ( f , b ) | : M 0 = cl . ( M \ S n × D n + 1 ) X 0 , X = X 0 D 2 n + 1 .
Given a presentation (e, a) of (f, b) as in (i) define a presentation of (f′, b′)
( e , a ) = ( g , c ) ( e , a ) : ( W ; M , M ^ ) = ( N ; M , M ) ( W ; M , M ^ ) X × ( I ; { 0 } , { 1 } ) .
                   THE ODD-DIMENSIONAL SURGERY OBSTRUCTION
The corresponding kernel split (−1)n-quadratic formation (F′, G′) for (f′, b′) is the effect of an algebraic surgery on the kernel (F, G) in (i) with data (H, χ, j: F → H *) such that
H = K n + 1 ( N , M ) = [ π 1 ( X ) ] , j * ( H ) = < x 1 , x 2 , , x > coker ( δ : G F * ) = K n ( M ) , μ W = ( θ 0 j γ χ ) : K n + 1 ( W ) = G H K n + 1 ( W ) * = G * H * ,
(p.332) with exact sequences of Z[π1(X)]-modules,
0 K n + 1 ( N ) K n + 1 ( W ) = G H ( δ ( 1 ) n + 1 j * ) K n + 1 ( W , N ) = K n + 1 ( W , M ) = F * K n ( N ) 0 , 0 K n + 1 ( M ) K n + 1 ( W ) = G H ( δ ( 1 ) n + 1 j * j γ χ + ( 1 ) n + 1 χ * ) K n + 1 ( W , M ) = F * H * K n ( M ) 0 , 0 K n + 1 ( M 0 ) K n + 1 ( W ) = G ( δ j γ ) K n + 1 ( W , M ) = F * H * K n ( M 0 ) 0 ,
and a commutative braid of exact sequences:
                   THE ODD-DIMENSIONAL SURGERY OBSTRUCTION
The normal map (e, a): NX × I is (n + 1)-connected if and only if the Z[π1(X)]-module morphism,
( δ ( 1 ) n + 1 j * ) : G H F * ,
is onto, in which case K n+1(N) is a stably f.g. free Z[π1(X)]-module and the kernel (−1)n+1-quadratic form is given by
μ N = ( θ 0 j γ χ ) | : K n + 1 ( N ) = ker ( ( δ ( 1 ) n + 1 j * ) : G H F * ) K n + 1 ( N ) * .

(iv) Suppose given a presentation (e, a) of (f, b) as in (i) and data (H, χ, j) for algebraic surgery on the kernel split (−1)n-quadratic formation (F, G) with (p.333) H = Z[π1(X)]l and effect (F′, G′). Let x 1,x 2,…,x lK n(M) be the images of the basis elements (0,…, 0, 1, 0, …, 0) ∈ H under the composite

H j * F * = K n + 1 ( W , M ) K n ( M ) .
Use the b-framing section s b f r : I n + 1 ( f ) I n + 1 f r ( f ) (10.14) to identify each
x i K n ( M ) = s b ( I n + 1 ( f ) ) I n + 1 f r ( f )
with a regular homotopy class of framed n-immersions in (f, b). As in the proof of Proposition 10.25, x i contains framed n-embeddings. For n ⩾ 2 the framed n-embeddings can be varied within the regular homotopy class by arbitrary elements of Q ( 1 ) n + 1 ( [ π 1 ( X ) ] ) , as in the proof of Proposition 11.42. It is, therefore, possible to kill x 1, x 2, …, x lK n(M) by n-surgeries on (f, b) such that (F′, G′) is the kernel split (−1)n-quadratic formation for (f′, b′) obtained as in (iii).

(v) The Z[π1(X)]-module morphism

K n + 1 ( M ) K n + 1 ( N , M ) = H * = [ π 1 ( X ) ] ; y ( λ ( x 1 , y ) , λ ( x 2 , y ) , , λ ( x , y ) )
is onto, so that the braid in (iii) identifies K n(M′) with the cokernel of the Z[π1(X)]-module morphism
K n + 1 ( N , M ) = H = [ π 1 ( X ) ] K n ( M ) = K n ( M 0 ) ; ( a 1 , a 2 , a ) a 1 x 1 + a 2 x 2 + + a x .
 □

Remark 12.37 For any normal bordism between n-connected (2n+1)-dimensional normal maps (f, b), (f′, b′),

( ( g , c ) ; ( f , b ) , ( f , b ) ) : ( N 2 n + 2 ; M 2 n + 1 , M 2 n + 1 ) X × ( I ; { 0 } , { 1 } ) ,
it is possible to kill the kernel Z[π1(X)]-modules K i(N) (in) by surgery below the middle dimension. Thus (g, c): NX × I can be made (n + 1)-connected, with K n+1(N, M) a f.g. free Z[π1(X)]-module of rank l ⩾ 0 (say). The cobordism (N; M, M′) is the trace of l n-surgeries on (f, b) with geometric effect (f′, b′), and with algebraic effect given by 12.36. □

Algebraic surgery on formations can also be used for purely algebraic computations:

Proposition 12.38 Let A be a principal ideal domain with involution, with quotient field K. (p.334)

  1. (i) Every split ε-quadratic formation (F, G) over A is cobordant to a formation (F′,G′) with δ′: G′ → F′ * injective.

  2. (ii) L 2n+1(K) = 0.

Proof (i) The A-module coker(δ: GF *) is finitely generated, so that it can be expressed as a direct sum of a f.g. free A-module S and a f.g. torsion A-module T:

coker ( δ ) = S T .
(Here, torsion means that aT = 0 for some a ≠ 0 ∈ A.) The sesquilinear pairing,
λ : coker ( δ ) × ker ( δ ) A ; ( x , y ) x ( γ ( y ) ) ,
is such that for any x ∈ S, there exists y ∈ ker(δ) such that λ(x,y) = 1 ∈ A. The abstract version of 12.36 (iv) shows that for any algebraic surgery on (F,G) with data (H, X, j) such that
j * : H = A coker ( δ ) ; ( a 1 , a 2 , a ) a 1 x 1 + a 2 x 2 + + a x
the effect of the algebraic surgery (F′,G′) has δ′: G′ → F′ * injective with
coker ( δ ) = T .

(ii) Take A = K, ∈ = (−1)n in (i). The only torsion A-module is 0, so δ′ is an isomorphism, and (F′, G′) is a trivial split (−1)n-quadratic formation. □

12.6 Linking forms

Odd-dimensional surgery obstructions were originally formulated by Kervaire and Milnor [38] and Wall [90] in terms of linking forms, but the method only applies to finite fundamental groups π. However, linking forms remain useful tools in surgery theory. This section is a brief introduction to linking forms and their use in the computation L 2*+1(Z) = 0. See Chapter 3 of Ranicki [70] for a considerably more complete account.

Let A be a ring with involution, and let SA be a subset such that

  1. (i) each sS is a central non-zero divisor, with S,

  2. (ii) if s, tS then stS,

  3. (iii) 1 ∈ S.

Definition 12.39 (i) The localization of A is the ring of fractions

S 1 A = A × S / { ( a , s ) ˜ ( b , t ) | at = bs A } ,
with elements denoted α s . The natural map
A S 1 A ; a a 1
is an injective morphism of rings with involution.

(p.335) (ii) An (A, S)-module T is a f.g. A-module such that sT = 0 for some sS, and which admits a f.g. free A-module resolution of the type

0 F 1 d F 0 T 0.
The dual of T = coker(d) is the (A, S)-module
T ^ = Hom A ( T , S 1 A A ) = coker ( d * : ( F 0 ) * ( F 1 ) * ) .

(iii) A nonsingular ε-symmetric linking form (T, λ) over (A, S) is an (A, S)- module T together with a sesquilinear pairing

λ : T × T S 1 A A
such that for all x, y, zT; a, bA
  1. (a) λ(x, y + z) = λ(x, y) + λ(x, z) ∈ S −1 A/A,

  2. (b) λ(ax, by) = (x, y)āS −1 A/A,

  3. (c) λ ( y , x ) = λ ( x , y ) ¯ S 1 A / A ,

  4. (d) the adjoint A-module morphism

λ : T T ^ ; x ( y λ ( x , y ) )
is an A-module isomorphism.

(iv) A nonsingular split ε-quadratic linking form (T, λ, v) over (A, S) is an ε-quadratic linking form (T, λ) together with a function

v : T Q ( S 1 A A ) = ( S 1 A A ) / { b b ¯ | b S 1 A A }
such that
  1. (a) ν(ax)=ā∈Q (S −1 A/A)(x∈T,a∈A),

  2. (b) ν(x+y)−ν(x)−ν(y)=λ(x,y)∈Q (S −1 A/A)(x,y∈T)

  3. (c) λ ( x , x ) = v ( x ) + v ( x ) ¯ S 1 A A ( x T ) .

 □

Example 12.40 Let A be an integral domain, and let S = A\{0} ⊂ A.

  1. (i) The localization S −1 A = K is the quotient field of A.

  2. (ii) If A is a principal ideal domain an (A, S)-module T is a f.g. A-module such that sT = 0 for some sS. For any f.g. A-module H the torsion submodule THH is an (A, S)-module, and H/TH is a f.g. free A-module. □

There is a close connection between formations and linking forms:

Proposition 12.41 The isomorphism classes of nonsingular split ε-quadratic linking forms (T, λ, v) over (A, S) are in one–one correspondence with the stable (p.336) isomorphism classes of split (−∈)-quadratic formations ( F , ( ( γ δ ) , θ ) G ) over A such that δ: S –1 GS −1 F * is an S −1 A-module isomorphism.

Proof Given such a split formation (F, G) define a linking form (T, λ, v) by

T = coker ( δ : G F * ) , λ : T × T S 1 A A ; ( [ x ] , [ y ] ) y ( γ ( z ) ) s , v : T Q ( S 1 A A ) ; [ x ] θ ( z ) ( z ) s s ¯ , ( x , y F * , z